Thursday, September 23, 2010

Theory Update 4

Young twistor particle physicists are turning up everywhere these days. Mathew Bullimore has a recent paper on twistor diagram techniques, which is well worth reading. New papers by Mason, Skinner, Arkani-Hamed, Cachazo and others are continually appearing. We still await the official appearance of associahedra in a description of these structures, but at least the concept of information dimension is now thoroughly entrenched at the heart of modern scattering techniques.


  1. The article of Bullimore discusses technical details which are not easy to understand from the recent article of Nima and others and is of help for a beginner like me.

  2. Twistor geometry seems to emerge in a couple of ways in what I am doing. They emerge as E_6 Jordan systems. G_2 though is the automorphism of the Jordan group, as F_4 and G_2 are centralizer of E_8. The extension to the E_6 valued Jordan matrix leads to twistor space structures and that the norm of states obeys

    (φ|ψ) = (2ħ)^{-1}{Ω(φ,ψ) + iΩ(φ,gψ)],

    where g is a group operation and Ω the symplectic operation. Parentheses are used for bra-ket operations because this editor does not like carrot symbols. The group operation is g_2, which constructs the E_6 Jordan algebra from the Hermitian and anti-Hermitian Jordan matrices.

    Twistor space is also a manifestation of Klein group structure on hyperbolic manifolds. Klein group construct light cones by the identification with poles on S^n (accumulation points), but in a hyperbolic setting. The null condition on light rays is defined by a projective geometry or equivalently by a stabilizer of O(2,n) (elements g such that gx = x) that is


    where Hei(2n-3) is the Heisenberg group of 2n – 3 dimensions. In AdS_{n+1} with two timelike directions the projective geometry is then “mod 2-planes = P,” just as regular projective space is R^n/x ~ λx and the lightcones or null directions are defined by O(2,n)/P. Now if we include the positive and negative sequences on S^n, this then constructs the space PT^+ and PT^- in twistor theory.
    There are connections between string theory and twistor space as found by Witten

    and for the AdS_5 spacetime this twistor space is E_6. This is where the Weyl curvature enters into the picture. This twistor spacetime is equivalent to perturbative conformal theory, which again by AdS/CFT gives a perturbation theory of E_n according to isometries --- the Weyl curvature.

  3. Connections between G_2 and twistors seem plausible also from TGD point of view. My earlier approach has been strongly number theoretic. After reading the article of Nima et al I have been working with the generalization of twistor string theory to TGD framework.

    One can say that both are almost topological QFT:s. Twistorialization replaces points of M^4 with spheres in twistor space CP_3. The replacement of point with 2-D parton means that 2 additional dimensions are added so that you have 6-D surfaces in some space. This space turns to be the product of twistor space and its dual with metrics with 2,4 signature and Euclidian signature. The conjecture is that Calabi-Yau structure generalizes also to 2,4 signature.

    The almost trivial observation that I should made for long time ago is that the pairs of points of CP_3 can be mapped to both M^4 point and CP_2 as a plane in CP_3 and the maps commute. Therefore the twistor pairs are mapped naturally to M^4xCP_2.

    The space-time surfaces correspond to 6-D holomorphic surfaces in CP_3xCP_3 replacing Witten's twistor strings. I ended up with an explicit proposal for the conditions guaranteing that the field equations for the preferred extremals of Kaehler action are satisfied. This comes just from general principles (general coordinate invariance, effective 2-dimensionality, the fact that twistor space is Calabi Yau, and the fact that only the holomorphic 3-forms associated with Calabi-Yau's can appear in the second order partial differential equations needed) and from the extremely restrictive condition of holomorphy. Infinite families of obvious solutions exist. It would be fantastic if this would be the long sought for explicit form for the solutions of field equations. Probably too much to hope!

    If the identification of space-time surfaces as hyper-quaternionic surfaces of M^4xCP_2 and as lifts to the holomorphic surfaces of CP_3xCP_3 are equivalent this means deep connections between the number theoretic and twistor approaches. SU(3) would appear in many roles: as subgroup of automorphisms of octonions, as holonomy group of twistors, and as color group. Geometry, number theory, physics!

    For details see my blog and pdf articles referred there.

  4. Matti,

    I can connect up with the CP^3 construction, which is a natural form of twistor space. I will confess that I have trouble seeing the crux of what it is you work on.

    The equation I wrote above when it acts on a spinor valued quantity σ^A

    = (2ħ)^{-1}[Ω(φ,σ^Aψ) + iΩ(φ,g(σ^Aψ))]

    is a twistor equation, where the group element g acts as the momentum "boost" in

    ω’^A = ω^A +iq^{AA’}π_{A’}.

    Time got a bit late again, so I will have to try again later.

  5. I did not mean any detailed connection to you work about which I probably understand as little as you about the work of mine on basis of this kind short communication;-), just the general idea that classical number fields and twistors seem to be related.

    The general problem of blog discussions is that people have only vague impressions about what others are saying.

  6. What a comfort to hear! I have felt so stupid :) But even two guys working on the same thing don't understand each other, piuh...


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